Vector Dot Product and Cross Product
DOT PRODUCT
The dot product (also called the scalar product) of 2 two-dimensional vectors \(\overline{u} = <u_1,u_2>\) and \(\overline{v} = <v_1,v_2>\) is defined as \[ \overline{u} \cdot \overline{v} = u_1v_1 + u_2v_2.\] In other words, we multiply the 1st components of the two vectors together and add it to what we get when we multiply the 2nd components of the two vectors together.
For three-dimensional vectors \(\overline{u} = <u_1,u_2,u_3>\) and \(\overline{v} = <v_1,v_2,v_3>\), the dot product is defined as \[ \overline{u} \cdot \overline{v} = u_1v_1 + u_2v_2 + u_3v_3.\] The resulting quantity is a scalar.
Example: If \(\overline{u} = (3,5)\) and \(\overline{v} = (-1,2)\), find \(\overline{u} \cdot \overline{v}\).
Solution: \[\begin{equation}
\begin{split}
\overline{u} \cdot \overline{v} & = (3,5) \cdot (-1,2)\\
& = (3)(-1) + (5)(2) \\
& = 7\\
\end{split}
\end{equation}\]
GRAPHICAL INTERPRETATION
The dot product of 2 vectors, their lengths, and the angle between them are all related through the equation
\[\overline{u} \cdot \overline{v} =|u||v|\cos{\theta}.\] Here, \(|\overline{u}|\) denotes the magnitude (length) of the vector \(\overline{u}\), \(|\overline{v}|\) denotes the magnitude (length) of the vector \(\overline{v}\) and \(\theta\) is the angle between the vectors \(\overline{u}\) and \(\overline{v}\). Of course, we could also rearrange this formula to solve for the angle \(\theta\) between any 2 vectors:
\[\cos{\theta} = \frac{\overline{u} \cdot \overline{v}}{|\overline{u}||\overline{v}|}.\] Notice that a dot product of 0 means that two vectors are perpendicular.
CROSS PRODUCT
For 2 three-dimensional vectors \(\overline{u} = (u_1,u_2,u_3)\) and \(\overline{v} = (v_1,v_2,v_3)\), the cross product is defined as:
\[ \overline{u} \times \overline{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1).\]
Note: The cross product \(\overline{u} \times \overline{v}\) is only defined for 2 three-dimensional vectors, and it produces a vector that is perpendicular to both \(u\) and \(v\).
Aside: For those that have taken a course in linear algebra and are familiar with the concept of “determinants,” you can also define the cross product as
\[
\overline{u} \times \overline{v} =
\left(
\begin{vmatrix}
u_2 & u_3 \\
v_2 & v_3 \\
\end{vmatrix}
,
-
\begin{vmatrix}
u_1 & u_3 \\
v_1 & v_3 \\
\end{vmatrix}
,
\begin{vmatrix}
u_1 & u_2 \\
v_1 & v_2 \\
\end{vmatrix}
\right)
\]
Example: If \(\overline{u} = (2,0,5)\) and \(\overline{v} = (-1,1,3)\), find \(\overline{u} \times \overline{v}\).
Solution: \[\begin{equation}
\begin{split}
\overline{u} = (2,0,5) & = (2,0,5) \times (-1,1,3)\\
& = ((0)(3) - (5)(1), (5)(-1) - (2)(3),(2)(1) - (0)(-1)) \\
& = (0 - 5, -5 - 6, 2 - 0)\\
& = (-5 , -11, 2)\\
\end{split}
\end{equation}\]
